Investigating the Relationship between GDP per Capita and Other Socio-Economic Factors

Introduction:

For this data memo I have decided to investigate the relationship between GDP, Population Density, Fertility Rate, Life Expectancy, Under-5 Mortality Rate, and the Ratio of Population aged 0-14 and 65+ to those aged 15-64. This data memo consists of the following parts
```
 Part One: Regressing GDP on the other Variables

 Part Two: Regressing GDP on Variables with a quadratic term

 Part Three: Graphing Visuals

 Part Four: Reversing regressions to determine if endogeniety exists

 Part Five: Implementing Instrumental Variables
```

##Part One:

GDP = β0 + β1 x Population Density + β2 x Fertility Rate + β3 x Life Expectancy + β4 x Age Distribution

Statistical models
	Model 1
(Intercept)	-252114.782^***
	(34468.855)
`POP Dens`	3.744^***
	(0.641)
Fert	-7427.944^*
	(3082.763)
Life	3204.198^***
	(432.904)
Mort	354.815^**
	(118.911)
`Youth and Old Age (0-14 and 65+)`	645.368^***
	(169.673)
R²	0.550
Adj. R²	0.540
Num. obs.	222
^*p < 0.001; ^p < 0.01; ^*p < 0.05

##Part Two: Investigating if there is a non-linear effect

GDP = β0 + β1 x Population Density + β2 x Fertility Rate + β3 x Fertility Rate squared + β4 x Life Expectancy + β5 x Life Expectancy Squared + β6 x Mortality <5 Rate + β7 Mortality Rate <5 Squared + β8 x Age Distribution

Statistical models
	Model 1
(Intercept)	1204515.630^***
	(200115.025)
`POP Dens`	2.391^***
	(0.609)
Fert	6571.272
	(6380.420)
Fert_squared	-1034.345
	(881.777)
Life	-35263.983^***
	(5293.902)
Life_squared	254.228^***
	(35.329)
Mort	15.147
	(208.006)
Mort_squared	-2.564
	(1.549)
`Youth and Old Age (0-14 and 65+)`	182.440
	(163.189)
R²	0.653
Adj. R²	0.640
Num. obs.	222
^*p < 0.001; ^p < 0.01; ^*p < 0.05

Drop insignificant and unnecessary terms.

GDP = β0 + β1 x Population Density + β2 x Life Expectancy + β3 x Life Expectancy Squared + β4 x Mortality <5 Rate + β5 Mortality Rate <5 Squared + β6 x Age Distribution

Statistical models
	Model 1
(Intercept)	1189825.196^***
	(192288.854)
`POP Dens`	2.390^***
	(0.590)
Life	-34486.721^***
	(5004.561)
Life_squared	248.228^***
	(32.790)
Mort_squared	-2.561^**
	(0.918)
`Youth and Old Age (0-14 and 65+)`	171.655
	(100.130)
R²	0.650
Adj. R²	0.642
Num. obs.	222
^*p < 0.001; ^p < 0.01; ^*p < 0.05

Part Three: Visuals

These analyses shows the effect of Life expectancy, Fertility, Mortality, Population Density, and the ratio of working-age to non-working-age on GDP per Capita, finding that these variables are all statistically significant. Furthermore, we find that the relationship between many of these variables are non linear, specifically those relationships between GDP per Capita, Under 5 mortality Rate, and Life Expectancy,

However, when the model is adjusted for quadratic terms, neither Fertility nor Fertility squared are statistically significant. One aspect that should be considered is the effect of GDP per Capita on the independent variables. To test for this, I will reverse the regression to see if endogeniety exists.

Part 4 Reversing the Regressions

population Density = β0 + β1 x GDP Per Capita + β2 x Fertility Rate + β3 x Life Expectancy + β4 x Mortality <5 Rate + β5 x Age Distribution

Statistical models
	Model 1
(Intercept)	-3444.437
	(3791.646)
GDP_per_capita	0.036^***
	(0.006)
Fert	-221.447
	(307.863)
Life	40.708
	(47.745)
Mort	21.509
	(11.882)
`Youth and Old Age (0-14 and 65+)`	2.925
	(17.292)
R²	0.244
Adj. R²	0.226
Num. obs.	222
^*p < 0.001; ^p < 0.01; ^*p < 0.05

Fertility Rate = β0 + β1 x GDP Per Capita + β2 x population Density + β3 x Life Expectancy + β4 x Mortality <5 Rate + β5 x Age Distribution

Statistical models
	Model 1
(Intercept)	1.036
	(0.836)
GDP_per_capita	-0.000^*
	(0.000)
`POP Dens`	-0.000
	(0.000)
Life	-0.018
	(0.010)
Mort	0.014^***
	(0.002)
`Youth and Old Age (0-14 and 65+)`	0.041^***
	(0.003)
R²	0.866
Adj. R²	0.863
Num. obs.	222
^*p < 0.001; ^p < 0.01; ^*p < 0.05

Life Expectancy = β0 + β1 x GDP Per Capita + β2 x Population Density + β3 x Fertility Rate + β4 x Mortality <5 Rate + β5 x Age Distribution

Statistical models
	Model 1
(Intercept)	78.481^***
	(0.834)
GDP_per_capita	0.000^***
	(0.000)
`POP Dens`	0.000
	(0.000)
Fert	-0.730
	(0.436)
Mort	-0.178^***
	(0.012)
`Youth and Old Age (0-14 and 65+)`	0.028
	(0.025)
R²	0.861
Adj. R²	0.858
Num. obs.	222
^*p < 0.001; ^p < 0.01; ^*p < 0.05

Mortality <5 Rate = β0 + β1 x GDP Per Capita + β2 x population Density + β3 x Life Expectancy + β4 x Fertility Rate + β5 x Age Distribution

Statistical models
	Model 1
(Intercept)	216.762^***
	(15.767)
GDP_per_capita	0.000^**
	(0.000)
`POP Dens`	0.001
	(0.000)
Life	-2.839^***
	(0.191)
Fert	9.386^***
	(1.631)
`Youth and Old Age (0-14 and 65+)`	-0.054
	(0.098)
R²	0.864
Adj. R²	0.861
Num. obs.	222
^*p < 0.001; ^p < 0.01; ^*p < 0.05

The results from reversing these regressions shows that GDP per Capita and each variable have reciprocal effects on one another. Therefore, it is hard to tell which variable is the cause and which variable is the effect. This is intuitive. Since, on one hand, if a country has a low GDP per Capita, then their citizens are less likely to have access to quality medical care. The lack of access to quality medical care decreases their life expectancy and fertility rate while increasing their Under 5 Mortality Rate. On the other hand, a country with a high life expectancy means more people are working for longer, with fewer worries and more focus on education and employment.The increase in education and employment then increases a countries GDP per Capita.

Part 5: Instrumental Variables

Perhaps the most challenging aspect of using instrumental variables is finding one that will work. This is because finding a variable that doesn’t affect your dependent variable, but still affects your independent variable is extremely challenging. However, in my case, a good predictor is right in front of me. In every regression I ran, population density was statistically significant at less than a 5% significance level (in many cases near 0 p-value). Furthermore, population density was not a statistically significant variable for the life expectancy regression. So I will use this variable to determine the effect of GDP per Capita on life expectancy.

Life = β0 + β1 x GDP_per_capita + β2 x (GDP_per_capita)^2 + β3 x Fertility Rate + β4 x Mortality Rate + β5 x Youth and Old Age (0-14 and 65+), where Population Density is used as an instrumental variable for GDP_per_capita.

Model Coefficients
	Term	Estimate	Std..Error	t.value	Pr…t..
(Intercept)	(Intercept)	72.6094719	0.6195587	117.195471	0.0e+00
GDP_per_capita	GDP_per_capita	0.0001321	0.0000283	4.673629	5.2e-06

Model Coefficients

Model Coefficients
	Term	Estimate	Std..Error	t.value	Pr…t..
(Intercept)	(Intercept)	72.6094719	0.6195587	117.195471	0.0e+00
GDP_per_capita	GDP_per_capita	0.0001321	0.0000283	4.673629	5.2e-06

Diagnostics Tests

Diagnostics Tests
Test	df1	df2	Statistic	p.value
Weak instruments (GDP_per_capita)	1	220	65.492	3.93e-14
Weak instruments (GDP_per_capita^2)	1	220	87.924	< 2e-16
Weak instruments (Fert)	1	220	4.861	0.0285
Weak instruments (Mort)	1	220	3.513	0.0622
Weak instruments (Youth and Old Age)	1	220	0.097	0.7558
Wu-Hausman	1	215	4.444	0.0362
Sargan	NA	NA	NA	NA

The Diagnostic test statistics are instrumental to the assessment of the underlying validity and reliability of this model, they tell us specifically whether the use of population density as an instrumental variable for GDP per Capita is a valid approach. The p-value for both GDP per Capita terms is practically 0 suggesting that the use of population density and GDP per Capita is a valid approach.

Conclusion:

In conclusion, we find that GDP per Capita affects life expectancy and life expectancy affects GDP per Capita. We also discovered that this relationship is quadratic. The former discovery is called endogeneity and leads to biased coefficients; the latter is an interesting piece that will be used in the test. We then regressed GDP per Capita on life expectancy using population density as an instrumental variable. Following this, we conducted a diagnostic test. This test resulted in statistics that demonstrate the validity of the instrumental variable approach to account for endogeniety. However, it is important to consider that the variables population density and life expectancy may have correlation. If so then our chosen instrumental variable is flawed.

Altogether, we can say that, all else equal, a $10,000 increase in a country’s GDP per Capita is associated with a 1.32-year increase in the average life expectancy of their population. Furthermore, the coefficient and significance level of GDP per Capita squared tell us that this effect is larger for low-income countries and smaller for high-income countries.

Data Memo 3

Maurizio Campana

2024-04-1